Linking object data but not the key frames
Suggestion.
Know some features of Follow Path Constraint. The constraint ..


active or inactive with keyframes ... see eye icon
[influence] can be 0 to 1.0 with keyframes
You can stack up many Path Constraints on a single object with judiciously placed keyframes.  Like relay runners passing a baton.
Thus the object can travel down each path one at a time with no influence from the other.  You known when you place paths near each other the movement may seem continous


I suggest you use these 3 features to address your issues.